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Field and reverse field solitons in wave-operator nonlinear Schrödinger equation with space-time reverse: Modulation instability

  • H I Abdel-Gawad
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  • Mathematics Department Faculty of Science Cairo University, Giza, Egypt

Received date: 2022-12-27

  Revised date: 2023-04-09

  Accepted date: 2023-04-19

  Online published: 2023-06-07

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© 2023 Institute of Theoretical Physics CAS, Chinese Physical Society and IOP Publishing

Abstract

The wave-operator nonlinear Schrödinger equation was introduced in the literature. Further, nonlocal space–time reverse complex field equations were also recently introduced. Studies in this area were focused on employing the inverse scattering method and Darboux transformation. Here, we present an approach to find the solutions of the wave-operator nonlinear Schrödinger equation with space and time reverse (W-O-NLSE-STR). It is based on implementing the unified method together with introducing a conventional formulation of the solutions. Indeed, a field and a reverse field may be generated. So, for deriving the solutions of W-O-NLSE-STR, it is evident to distinguish two cases (when the field and its reverse are interactive or not-interactive). In the non-interactive and interactive cases, exact and approximate solutions are obtained. In both cases, the solutions are evaluated numerically and they are displayed graphically. It is observed that the field exhibits solitons propagating essentially (or mainly) on the negative space variable, while those of the reverse field propagate on the other side (or vice versa). These results are completely novel, and we think that it is an essential behavior that characterizes a complex field system with STR. On the other hand, they may exhibit right and left cable patterns (or vice versa). It is found that the solutions of the field and its reverse exhibit self-phase modulation by solitary waves. In the interactive case, the pulses of the field and its reverse propagate in the whole space. The analysis of modulation stability shows that, when the field is stable, its reverse is unstable or both are stable. This holds whenever the polarization of the medium is self-defocusing.

Cite this article

H I Abdel-Gawad . Field and reverse field solitons in wave-operator nonlinear Schrödinger equation with space-time reverse: Modulation instability[J]. Communications in Theoretical Physics, 2023 , 75(6) : 065005 . DOI: 10.1088/1572-9494/acce32

1. Introduction

The nonlinear Schrödinger equation (NLSE) with space-time reverse (STR) was very recently considered in the literature. In this context, the inverse scattering transform (IST) for the STR- NLSE with nonzero boundary conditions at infinity was presented in [1, 2]. The IST for rapidly decaying data was constructed for nonlocal reverse space-time Sine/Sinh-Gordon type equations [2, 3]. The nonlocal STR- parity-time (PT)-symmetric-multi-component NLSEs under a specific nonlocal group reduction was studied in [4]. Therein generation of their ISTs and soliton solutions were carried via the Riemann-Hilbert technique. An STR nonlocal Sasa–Satsuma equation was introduced and its solutions with the binary Darboux transformation (DT) method were derived in [5]. The IST for nonlocal complex reverse-space-time multicomponent integrable modified Korteweg–de Vries (mKdV) equations was considered in [6]. Bright and dark soliton solutions to the partial STR nonlocal Mel’nikov equation with PT-symmetry were constructed by the Hirota bilinear method (HBM) and with the Kadomtsev–Petviashvili (KP) hierarchy reduction method [7]. These are simple approaches to model potentially negative-value stationary space–time models, and for such models, it was shown that they possess a reverse (pointing upward) dimple [8]. An STR Fokas–Lenells (FL) equation was derived from a rather simple, but extremely important symmetry reduction of the corresponding local equation [9]. In [10], a general coupled integrable dispersionless system and nontrivial solutions in terms of the ratio of determinants were obtained using matrix DT. Reverse space and/or time nonlocal FL equations with higher-order nonlinear effects were considered via HBM in [11]. Multiple soliton solutions for the STR m KdV equation were found, which were classified to special soliton solutions, with explicit 1-soliton and 2-soliton [12]. In [13], the DT for the STR derivative NLSE was constructed. Therein, breathers and rogue waves on the double-periodic background were inspected by DT by using a plane wave seed solution. The nonlocal complex STR modified Korteweg–de Vries (mKdV) hierarchies via nonlocal symmetry reductions of matrix spectral problems was explored with their soliton solutions by the IST in [14]. Exact periodic and localized solutions of a nonlocal STR- Mel’nikov equation were derived by the HBM [15]. In [16], time reverse modeling was applied to localize and characterize acoustic emission using a numerical concrete model with a method for exploration of geophysics to non-destructive testing. An integrable STR nonlocal Sasa–Satsuma equation was introduced and the DT was used to obtain the soliton solutions [17].
The wave-operator is currently considered in quantum theory. The wave operator was shown to be a solution of an operator equation which is the analog of the Moller equations of the scattering theory [18]. Here, our interest is on wave operator–complex field equations, particularly on nonlinear Schrödinger equations (WO-PNLSE). This was not considered in the literature, while many studies were carried out on WO-NLSE. Analytic studies of the behavior of solutions of WO-NLSE were rarely carried out. While, numerous numerical studies on WO-NLSE were achieved. In this context, conservative finite difference schemes for an initial-boundary value problem of WO-NLSE were presented [19]. In [20], the uniform error estimates of finite difference methods for the WO-NLSE with a perturbation strength described by a dimensionless parameter were determined. A linearized finite element method for solving the WO-NLSE was proposed, where a modified leap–frog scheme is applied for time discretization and a Galerkin finite element method is applied for spatial discretization were used [21, 22]. A fully discrete scheme by discretizing the space with the local discontinuous Galerkin method and the time with the Crank–Nicholson scheme to simulate the multi-dimensional WO-NLSE was proposed [23]. An exponential wave integrator sine pseudo spectral method for the WO-NLSE and rigorous error analysis was suggested [24]. Their in NLSE is considered perturbed by the wave operator via a small parameter so that WO-NLSE converges to NLSE. In [25], the initial-boundary value problem of a class of NLSE with wave operator was considered by implementing an efficient finite difference scheme. The Galerkin finite element method to numerically solve the nonlinear fractional Schrödinger equation with wave operator was employed [26]. In [27], the scattering theory for the coupled wave-Schrödinger equation with the Yukawa type interaction, which is a certain quadratic interaction in three space dimensions, was studied. Efficient solution of the NLSE with wave operator, subject to periodic boundary conditions was derived [28]. A fully discrete scheme by discretizing the space with the local discontinuous Galerkin method and the time with the Crank–Nicholson scheme to simulate the multi-dimensional Schrödinger equation with a wave operator was presented [29, 30]. Some relevant works in this area were carried out [3139]
Here, we consider the wave-operator nonlinear Schrödinger equation with space and time reverse (W-O-NLSE-STR) and use conventional forms for its solutions in two cases; when the field and its reverse are not interactive and when they are interactive. This is done together with implementing the unified method (UM) [4045]. The UM asserts that the solutions of nonlinear partial differential equations (NLPDEs) with constant coefficients are expressed in polynomial and rational forms in auxiliary functions with appropriate auxiliary equations (AEs). Here, we confine ourselves to find the rational solutions. It is worth mentioning that in the literature, the problems of complex field equation with STR were dealt with, only via the inverse scattering method and DT. The approach proposed here is straight forwardly applicable; thus, complicated complex field equations are directly amenable.
The outlines of this paper are as follows.
Section 2 is concerned with the mathematical formulation and a brief account of the UM. The solutions in the case when the field and its reverse are not interactive are presented in section 3. The solutions in the case when the field and its reverse are interactive are presented in section 4.
Section 5 is devoted to studying the modulation stability (MS). In section 6, conclusions are given.

2. Mathematical formulation and a brief account of the UM

2.1. Mathematical formulation

The W-O-NLSE with wave-operator reads,
$\begin{eqnarray}\begin{array}{l}{\rm{i}}\displaystyle \frac{\partial w(x,t)}{\partial t}\\ +\hat{W}w(x,t)+\beta | w(x,t){| }^{2}w(x,t)=0,\end{array}\end{eqnarray}$
where $\hat{W}w(x,t)={\gamma }^{2}\tfrac{{\partial }^{2}w(x,t)}{\partial {t}^{2}}-{\alpha }^{2}\tfrac{{\partial }^{2}w(x,t)}{\partial {x}^{2}},$ which accounts for the hyperbolic effect on the structure of pulses propagation in optical fibers, thus (1) becomes,
$\begin{eqnarray}\begin{array}{l}{\rm{i}}\displaystyle \frac{\partial w(x,t)}{\partial t}+{\gamma }^{2}\displaystyle \frac{{\partial }^{2}w(x,t)}{\partial {t}^{2}}-{\alpha }^{2}\displaystyle \frac{{\partial }^{2}w(x,t)}{\partial {x}^{2}}\\ \,+\,\beta | w(x,t){| }^{2}w(x,t)=0,\end{array}\end{eqnarray}$
where w(x, t) is the complex envelope field, γ is the measure the hyperbolic effect, α is the group dispersion velocity and β stands for self-focusing or self-defocusing when β > 0 or β < 0, respectively.
The W-O-NLSE-STR is,
$\begin{eqnarray}\begin{array}{l}{\rm{i}}\displaystyle \frac{\partial w(x,t)}{\partial t}+{\gamma }^{2}\displaystyle \frac{{\partial }^{2}w(x,t)}{\partial {t}^{2}}-{\alpha }^{2}\displaystyle \frac{{\partial }^{2}w(x,t)}{\partial {x}^{2}}\\ \,+\,\beta w{\left(x,t\right)}^{2}w(-x,-t)=0,\end{array}\end{eqnarray}$
where w(−x, − t) is the complex envelope reverse field.
Here, we search for solutions with complex amplitude. To this issue, we introduce the transformations,
$\begin{eqnarray}\begin{array}{l}w(x,t)=\left({u}_{1}(x,t)+{\rm{i}}{u}_{2}(x,t)\right){{\rm{e}}}^{{\rm{i}}({kx}-t\omega )},\\ w(-x,-t)=\left({u}_{1}(-x,-t)+{\rm{i}}{u}_{2}(-x,-t)\right){{\rm{e}}}^{-{\rm{i}}({kx}-t\omega )}.\end{array}\end{eqnarray}$
When introducing (4) into (3), the real and imaginary parts are given respectively by,
$\begin{eqnarray}\begin{array}{l}{\alpha }^{2}{k}^{2}{u}_{1}(x,t)+\beta {u}_{1}(-x,-t){u}_{1}(x,t){}^{2}\\ -2\beta {u}_{1}(x,t){u}_{2}(-x,-t){u}_{2}(x,t)\\ -{\gamma }^{2}{\omega }^{2}{u}_{1}(x,t)+\omega {u}_{1}(x,t)2{\alpha }^{2}{{ku}}_{2x}(x,t)\\ -{\alpha }^{2}{u}_{1{xx}}(x,t)-\beta {u}_{1}(-x,-t){u}_{2}(x,t){}^{2}\\ +2{\gamma }^{2}\omega {u}_{2t}(x,t)+{\gamma }^{2}{u}_{1{tt}}(x,t)-{u}_{2t}(x,t)=0,\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{\alpha }^{2}{k}^{2}{u}_{2}(x,t)+\beta {u}_{1}(x,t){}^{2}{u}_{2}(-x,-t)\\ -{\gamma }^{2}{\omega }^{2}{u}_{2}(x,t)+\omega {u}_{2}(x,t)\\ +2\beta {u}_{1}(-x,-t){u}_{1}(x,t){u}_{2}(x,t)\\ -\beta {u}_{2}(-x,-t){u}_{2}(x,t){}^{2}+{u}_{1t}(x,t)\\ -2{\gamma }^{2}\omega {u}_{1t}(x,t)+{\gamma }^{2}{u}_{2{tt}}(x,t)\\ -2{\alpha }^{2}{{ku}}_{1x}(x,t)-{\alpha }^{2}{u}_{2{xx}}(x,t)=0.\end{array}\end{eqnarray}$
We use the traveling waves transformations,
$\begin{eqnarray}\begin{array}{l}{u}_{1}(x,t)={U}_{1}(z),{u}_{2}(x,t)={U}_{2}(z),\\ {u}_{1}(-x,-t)={U}_{1}(-z),{u}_{2}(-x,-t)={U}_{2}(-z),\\ z={hx}+{rt}.\end{array}\end{eqnarray}$
By using (5), (6) and (7), respectively, gives rise to,
$\begin{eqnarray}\begin{array}{l}{U}_{1}(z)\left(-{\gamma }^{2}{\omega }^{2}+{\alpha }^{2}{k}^{2}-2\beta {U}_{2}(-z){U}_{2}(z)+\omega \right)\\ +\beta {U}_{1}(-z)\\ {U}_{1}(z){}^{2}-\beta {U}_{1}(-z){U}_{2}(z){}^{2}-\beta {U}_{1}(-z){U}_{2}(z){}^{2}\\ -{\alpha }^{2}{h}^{2}{U}_{1}^{\prime\prime} (z)\\ +2{\alpha }^{2}{{hkU}}_{2}^{\prime} (z)+{\gamma }^{2}{r}^{2}{U}_{1}^{\prime\prime} (z)\\ +\,2{\gamma }^{2}r\omega {U}_{2}^{\prime} (z)-{{rU}}_{2}^{\prime} (z)=0,\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{U}_{2}(z)\left(-{\gamma }^{2}{\omega }^{2}+{\alpha }^{2}{k}^{2}+\omega \right)+\beta {U}_{2}(-z){U}_{1}(z){}^{2}+2\beta {U}_{1}(-z)\\ {U}_{2}(z){U}_{1}(z)-{\alpha }^{2}{h}^{2}{U}_{2}^{\prime\prime} (z)-2{\alpha }^{2}{{hkU}}_{1}^{\prime} (z)+{\gamma }^{2}{r}^{2}{U}_{2}^{\prime\prime} (z)\\ -2{\gamma }^{2}r\omega {U}_{1}^{\prime} (z)+{{rU}}_{1}^{\prime} (z)-\beta {U}_{2}(-z){U}_{2}(z){}^{2}=0.\end{array}\end{eqnarray}$
Our objective is to find the exact and approximate analytic solutions of (8) and (9). So, the integrability of these equations is invoked. Recently it was established that a complex field equation is integrable when the real and imaginary parts are linearly dependent [37, 38].

2.2. Outline of the UM

Here, we are confined to obtain the rational solutions of (8) and (9). By using the integrability condition mentioned in the above, we write,
$\begin{eqnarray}\begin{array}{l}{U}_{2}(z)={{vU}}_{1}(z),\ {U}_{2}(-z)={{vU}}_{1}(-z),\\ {U}_{1}(z)=\displaystyle \frac{{\sum }_{j=0}^{j=s}{a}_{j}g{\left(z\right)}^{j}}{{\sum }_{j=0}^{j=s}{s}_{j}g{\left(z\right)}^{j}},\quad {U}_{1}(-z)=\displaystyle \frac{{\sum }_{j=0}^{j=s}{a}_{j}g{\left(-z\right)}^{j}}{{\sum }_{j=0}^{j=s}{s}_{j}g{\left(z\right)}^{j}},\\ {g}^{{\prime} }(z)=\displaystyle \sum _{j=0}^{j=k}{c}_{j}g{\left(z\right)}^{j},{g}^{{\prime} }(-z)=\displaystyle \sum _{j=0}^{j=k}{c}_{j}g{\left(-z\right)}^{j}.\end{array}\end{eqnarray}$
When substituting from (10) into (8) and (9) leads to the balance condition s = k − 1 or s = 2(k − 1),k > 1 and when k = 1, s = 2.
Comparison of the method used here with the known methods in the literature, we find the following. In the UM, exact solutions are found by inserting (10) into (8) and (9) and by stetting the coefficients $g{\left(z\right)}^{i},g{\left(-z\right)}^{j},i,j\,=\,0,1,2,\ldots {etc}$ equal to zero. In the case of finding approximate analytical solutions, some of these coefficients are not taken identically zero, but they are considered as errors (residue terms (RT)). The maximum error (ME) is controlled via an adequate choice of the parameters in the RT It is worth mentioning that the ME found here, is space and time independent. This is not the case when evaluating the solutions via different numerical techniques.
In this paper the UM [40] was used. It unifies all known methods such as, the tanh, modified [46], and extended versions, the F-expansion [47], the exponential, the G’/G expansion method [48], the Lie symmetries [49] and the Kerdyashov method [50]. On the other hand, in the applications, it is established that the UM [51, 52] is of low time cost in symbolic computations. Thus, we think that it prevails the use of the Lie group to construct the symmetries of NLPDEs, as it is required to carry a hierarchy of long steps. Furthermore, it provides a wide class of solutions which range from hyperbolic solutions, periodic solutions to elliptic solutions in Jacobi elliptic functions.

3. The non-interactive case

3.1. Exact solutions of (8) and (9)

3.1.1. When k = 2, s = 1

The solutions are expressed by,
$\begin{eqnarray}\begin{array}{l}{U}_{1}(z)=\displaystyle \frac{{a}_{1}g(z)+{a}_{0}}{{s}_{1}g(z)+{s}_{0}},\quad {U}_{2}(z)=\displaystyle \frac{v\left({a}_{1}g(z)+{a}_{0}\right)}{{s}_{1}g(z)+{s}_{0}},\\ {U}_{1}(-z)=\displaystyle \frac{{a}_{1}g(-z)+{a}_{0}}{{s}_{1}g(-z)+{s}_{0}},\quad {U}_{2}(-z)=\displaystyle \frac{v\left({a}_{1}g(-z)+{a}_{0}\right)}{{s}_{1}g(-z)+{s}_{0}},\\ g^{\prime} (z)={c}_{2}g{\left(z\right)}^{2}+{c}_{1}g(z)+{c}_{0},\\ g^{\prime} (-z)={c}_{2}g{\left(-z\right)}^{2}+{c}_{1}g(-z)+{c}_{0}.\end{array}\end{eqnarray}$
From (11) into (8) and (9) and by setting the coefficients of $g{\left(z\right)}^{i}$ and $g{\left(-z\right)}^{j}$, i, j = 0, 1, 2,.., equal to zero, leads to,
$\begin{eqnarray}\begin{array}{l}{s}_{0}=\displaystyle \frac{{a}_{0}{c}_{2}^{2}{s}_{1}\sqrt{{\gamma }^{2}{r}^{2}-{\alpha }^{2}{h}^{2}}-{c}_{2}\sqrt{9{a}_{0}^{2}{c}_{2}^{2}{s}_{1}^{2}\left({\alpha }^{2}{h}^{2}-{\gamma }^{2}{r}^{2}\right)-2{a}_{1}^{4}\beta \left({a}_{0}^{2}{c}_{2}^{2}{s}_{1}^{2}-1\right)}}{2{a}_{1}{c}_{2}^{2}\sqrt{{\alpha }^{2}{h}^{2}-{\gamma }^{2}{r}^{2}}},\\ \beta =-\displaystyle \frac{{s}_{1}^{2}\left(-{\gamma }^{2}{\omega }^{2}+{\alpha }^{2}{k}^{2}+\omega \right)}{{a}_{1}^{2}\left({v}^{2}+1\right)},\\ {c}_{0}-\displaystyle \frac{1}{4{c}_{2}{\left({\alpha }^{2}{h}^{2}-{\gamma }^{2}{r}^{2}\right)}^{2}}(\\ \left({\alpha }^{2}{h}^{2}-{\gamma }^{2}{r}^{2}\right)\left({c}_{1}^{2}\left({\alpha }^{2}{h}^{2}-{\gamma }^{2}{r}^{2}\right)-2\left(-{\gamma }^{2}{\omega }^{2}+{\alpha }^{2}{k}^{2}+\omega \right)\right)\\ -{v}^{2}{\left(-2{\alpha }^{2}{hk}-2{\gamma }^{2}r\omega +r\right)}^{2}),\\ \omega =\displaystyle \frac{{a}_{1}\left({c}_{1}\left({\gamma }^{2}{r}^{2}-{\alpha }^{2}{h}^{2}\right)+v\left(r-2{\alpha }^{2}{hk}\right)\right)+4{a}_{0}{c}_{2}\left({\gamma }^{2}{r}^{2}-{\alpha }^{2}{h}^{2}\right)}{2{a}_{1}{\gamma }^{2}{vr}},\\ {c}_{1}=\displaystyle \frac{16{a}_{0}^{2}{\gamma }^{2}{c}_{2}^{2}{r}^{2}\left({\alpha }^{2}{h}^{2}-{\gamma }^{2}{r}^{2}\right)+{a}_{1}^{2}\left({r}^{2}\left(4{\alpha }^{2}{\gamma }^{2}{k}^{2}+1\right)-4{\alpha }^{4}{h}^{2}{k}^{2}\right)}{16{a}_{0}{a}_{1}{\gamma }^{2}{c}_{2}{r}^{2}\left({\gamma }^{2}{r}^{2}-{\alpha }^{2}{h}^{2}\right)},\\ k=\displaystyle \frac{r\sqrt{48{a}_{0}^{2}{\gamma }^{2}{c}_{2}^{2}\left({\gamma }^{2}{r}^{2}-{\alpha }^{2}{h}^{2}\right)+{a}_{1}^{2}}}{2\alpha {a}_{1}\sqrt{{\alpha }^{2}{h}^{2}-{\gamma }^{2}{r}^{2}}}.\end{array}\end{eqnarray}$
By solving the AEs and by using (12) into (8) and (9), the solutions of (5) and (6) are,
$\begin{eqnarray}\begin{array}{l}{u}_{1}(x,t)=\displaystyle \frac{{P}_{1}}{{Q}_{1}},\\ {P}_{1}=2{a}_{0}{a}_{1}{c}_{2}^{2}\left({\alpha }^{2}{h}^{2}-{\gamma }^{2}{r}^{2}\right)\\ \quad \times \left(\sqrt{6}\tanh \left(\displaystyle \frac{\sqrt{6}{a}_{0}{c}_{2}\left({A}_{0}+z\right)}{{a}_{1}}\right)-3\right),\\ {Q}_{1}=2\sqrt{6}{a}_{0}{c}_{2}^{2}{s}_{1}\left({\alpha }^{2}{h}^{2}-{\gamma }^{2}{r}^{2}\right)\\ \quad \times \,\tanh \left(\displaystyle \frac{\sqrt{6}{a}_{0}{c}_{2}\left({A}_{0}+z\right)}{{a}_{1}}\right),\\ {u}_{2}(x,t)={{vu}}_{1}(x,t),\\ {u}_{1}(-x,-t)=\displaystyle \frac{{P}_{2}}{{Q}_{2}},\\ {P}_{2}=2{a}_{0}{a}_{1}{c}_{2}^{2}\left({\alpha }^{2}{h}^{2}-{\gamma }^{2}{r}^{2}\right)\\ \quad \times \left(\sqrt{6}\tanh \left(\displaystyle \frac{\sqrt{6}{a}_{0}{c}_{2}\left({A}_{0}-z\right)}{{a}_{1}}\right)-3\right)\\ {Q}_{2}=2\sqrt{6}{a}_{0}{c}_{2}^{2}{s}_{1}\left({\alpha }^{2}{h}^{2}-{\gamma }^{2}{r}^{2}\right)\tanh \left(\displaystyle \frac{\sqrt{6}{a}_{0}{c}_{2}\left({A}_{0}-z\right)}{{a}_{1}}\right),\\ {u}_{2}(-x,-t)={{vu}}_{1}(-x,-t),\quad z={hx}+{rt}.\end{array}\end{eqnarray}$
The results in (13) are evaluated numerically for the field and its reverse. The real parts,
$\begin{eqnarray}\begin{array}{rcl}{Rew}(x,t) & = & {u}_{1}(x,t)\cos ({kx}-t\omega )\\ & & -{u}_{2}(x,t)\mathrm{Sin}({kx}-t\omega ),\\ {Rew}(-x,-t) & = & {u}_{1}(-x,-t)\cos ({kx}-t\omega )\\ & & +{u}_{2}(-x,-t)\mathrm{Sin}({kx}-t\omega ),\end{array}\end{eqnarray}$
are displayed in figures 1(a)–(d).
Figure 1. (a)–(d). Rew(x, t) is displayed in figures 1(a) and (b), while Rew(−x, −t) is displayed in figures 1(c) and (d). When A0 = 0, c2 = 0.5, a1 = 9, α = 0.8, c2 = 1.4, a0 = 1.9, s1 = 2, r = 0.7, h = 1.2, v = 3.
Figures 1(a) and (b) show that the pulses of the field propagate on x < 0, while the values of the field is zero on x > 0.
Figures 1(c) and (d) show that the pulses of the reverse field propagate on x > 0, while the values of the reverse field is zero on x < 0.
When comparing 1(a), (c) and (b), (d), by varying the coefficient of the hyperbolic effect, γ, we observe that the pulses are periodic in time when γ = 0.5 while they are periodic in space when γ = 0.1. These figures exhibit gaps.

3.1.2. When k = 2, s = 2

In this case, we write,
$\begin{eqnarray*}\begin{array}{l}{U}_{1}(z)=\displaystyle \frac{{a}_{2}g{\left(z\right)}^{2}+{a}_{1}g(z)+{a}_{0}}{{s}_{2}g{\left(z\right)}^{2}+{s}_{1}g(z)+{s}_{0}},\quad {U}_{2}(z)={{vU}}_{1}(z),\\ {U}_{1}(-z)=\displaystyle \frac{{a}_{2}g{\left(-z\right)}^{2}+{a}_{1}g(-z)+{a}_{0}}{{s}_{2}g{\left(-z\right)}^{2}+{s}_{1}g(-z)+{s}_{0}},\end{array}\end{eqnarray*}$
$\begin{eqnarray}\begin{array}{l}{U}_{2}(-z)={{vU}}_{1}(-z),\\ g^{\prime} (z)]={c}_{2}g{\left(z\right)}^{2}+{c}_{1}g(z)+{c}_{0},\\ g^{\prime} (-z)={c}_{2}g{\left(-z\right)}^{2}+{c}_{1}g(-z)+{c}_{0}.\end{array}\end{eqnarray}$
When inserting (15) into (8) and (9), we have,
$\begin{eqnarray}\begin{array}{l}{s}_{1}=\displaystyle \frac{{c}_{2}{s}_{2}\left(-\sqrt{2{a}_{1}{a}_{2}{c}_{2}{c}_{1}{s}_{2}+{a}_{2}^{2}{c}_{1}^{2}+{a}_{1}^{2}{c}_{2}^{2}}+{a}_{2}{c}_{1}+{a}_{1}{c}_{2}\right)}{2{a}_{2}{c}_{2}^{2}},\\ \beta =\displaystyle \frac{{s}_{2}\left(2{a}_{0}{c}_{2}^{2}{s}_{2}\left({\alpha }^{2}{h}^{2}-{\gamma }^{2}{r}^{2}\right)-{a}_{2}\left(2{c}_{2}^{2}{s}_{0}\left({\alpha }^{2}{h}^{2}-{\gamma }^{2}{r}^{2}\right)\right)\right)}{{a}_{2}^{3}\left({v}^{2}+1\right)},\\ v=\displaystyle \frac{{a}_{1}\sqrt{{a}_{2}}{c}_{1}^{2}{c}_{2}\left({\gamma }^{2}{r}^{2}-{\alpha }^{2}{h}^{2}\right)-2\sqrt{{a}_{1}^{2}{a}_{2}{c}_{1}^{4}{c}_{2}^{2}{\left({\alpha }^{2}{h}^{2}-{\gamma }^{2}{r}^{2}\right)}^{2}}}{{a}_{1}\sqrt{{a}_{2}}{c}_{1}{c}_{2}\left(2\alpha h\sqrt{\omega }\sqrt{{\gamma }^{2}\omega -1}+r\left(2{\gamma }^{2}\omega -1\right)\right)}\\ {s}_{0}=0,\quad {c}_{0}=0,\quad k=\displaystyle \frac{\sqrt{\omega }\sqrt{{\gamma }^{2}\omega -1}}{\alpha }.\end{array}\end{eqnarray}$
By solving the AEs, and by using (16) into (15), we find the solutions of (5) and (6) are given by,
$\begin{eqnarray}\begin{array}{l}{u}_{1}(x,t)=\displaystyle \frac{{P}_{1}}{{Q}_{1}},\\ {P}_{1}={a}_{2}\left({a}_{2}{c}_{1}\left(3\tanh \left(\displaystyle \frac{3}{2}{c}_{1}\left({A}_{0}+z\right)\right)+1\right)-2{a}_{1}{c}_{2}\right),\\ {Q}_{1}={s}_{2}\left({a}_{1}\sqrt{{a}_{2}}{c}_{1}^{2}{c}_{2}{s}_{2}\left({\alpha }^{2}{h}^{2}-{\gamma }^{2}{r}^{2}\right)-{a}_{1}{c}_{2}\right.\\ {a}_{2}{c}_{1}\left(3\tanh \left(\displaystyle \frac{3}{2}{c}_{1}\left({A}_{0}+z\right)\right)+2\right)\\ \left.+\sqrt{2{a}_{1}{a}_{2}{c}_{2}{c}_{1}{s}_{2}+{a}_{2}^{2}{c}_{1}^{2}+{a}_{1}^{2}{c}_{2}^{2}}\right),\\ {u}_{2}(x,t)={{vu}}_{1}(x,t),\\ {u}_{1}(-x,-t)=\displaystyle \frac{{P}_{2}}{{Q}_{2}},\\ {P}_{2}={a}_{2}\left({a}_{2}{c}_{1}\left(3\tanh \left(\displaystyle \frac{3}{2}{c}_{1}\left({A}_{0}-z\right)\right)+1\right)-2{a}_{1}{c}_{2}\right),\\ {Q}_{1}={s}_{2}\left({a}_{1}\sqrt{{a}_{2}}{c}_{1}^{2}{c}_{2}{s}_{2}\left({\alpha }^{2}{h}^{2}-{\gamma }^{2}{r}^{2}\right)-{a}_{1}{c}_{2}\right.\\ {a}_{2}{c}_{1}\left(3\tanh \left(\displaystyle \frac{3}{2}{c}_{1}\left({A}_{0}-z\right)\right)+2\right)\\ \left.+\sqrt{2{a}_{1}{a}_{2}{c}_{2}{c}_{1}{s}_{2}+{a}_{2}^{2}{c}_{1}^{2}+{a}_{1}^{2}{c}_{2}^{2}}\right),\\ {u}_{2}(-x,-t)={{vu}}_{1}(-x,-t),\,z={hx}+{rt}.\end{array}\end{eqnarray}$
By using the results in (17) Rew(x, t) and Rew(−x, −t) are displayed in figures 2(a)–(f).
Figure 2. (a)–(f), Rew(x, t) is displayed in figures 2(a)–(c), while Rew(−x, −t) is displayed in figures 2(d) and (f). When &ohgr; = 5, γ = 2, α = 2, A0 = 0, c1 = 0.5, s2 = 6, s1 = 1.2, r = 0.5, h = 0.3, a2 = 4, a1 = 1.5, c2 = 1.5, a2 = 1.5.
Figures 2(a)–(c) show that the pulses of the field propagate mainly on x < 0, and they exhibit right cable shape, while those of the reverse field propagate mainly on x > 0, showing left cable shape. In both two fields self-phase modulation, by solitary waves, holds.

3.2. Approximate solutions of (8) and (9)

In section 3.1, we mentioned that the exact solutions of (8) and (9) are found by using the UM. The formal solutions, in g(z) and g(−z) (cf (11), are inserted into (8) and (9) and the coefficients of $g{\left(z\right)}^{i}$ and $g{\left(-z\right)}^{j}$, i, j = 0, 1, 2,.., are set equal to zero. In some cases, we find that the exact solutions are trivial. So, we try to get the approximate one’s by taking some RT non zero. Indeed, They are considered as errors. The ME is controlled by an adequate choice of the parameters in RT. This will be illustrated in what follows. On the other hand this represents a new prospective of using the UM.
Here, we consider the case k = 3, when taking s = 2 exact solutions are obtained. Unfortunately, the calculations are very lengthy and too complicated, so we take s = 1.
Thus, we consider (10) with the AEs are,
$\begin{eqnarray}\begin{array}{l}g^{\prime} (z)={c}_{3}g{\left(z\right)}^{3}+{c}_{2}g{\left(z\right)}^{2}+{c}_{1}g(z)+{c}_{0},\\ g^{\prime} (-z)={{\rm{c}}}_{3}{\rm{g}}{\left(-{\rm{z}}\right)}^{3}+{c}_{2}g{\left(-z\right)}^{2}+{c}_{1}g(-z)+{c}_{0}.\end{array}\end{eqnarray}$
From (11) and (18) into (8) and (9) gives rise to,
$\begin{eqnarray}\begin{array}{l}h=\displaystyle \frac{\gamma r}{\alpha },\quad k=\displaystyle \frac{1-2{\gamma }^{2}\omega }{2\alpha \gamma },\\ v=\displaystyle \frac{\sqrt{+4{a}_{1}^{2}\beta {\gamma }^{2}+{s}_{1}^{2}}}{2{a}_{1}\sqrt{-\beta }\gamma },\quad {a}_{0}=-\displaystyle \frac{{a}_{1}{s}_{0}}{{s}_{1}}.\end{array}\end{eqnarray}$
In (18), we remark that ci, i = 0, 1, 2, 3 are arbitrary. The RT are: $\left\{-\tfrac{{a}_{1}{s}_{0}^{3}}{2{\gamma }^{2}},\tfrac{{a}_{1}{s}_{0}{s}_{1}^{2}}{2{\gamma }^{2}}\right\}.$ This means that, here, the parameters a1, s1, s0 and γ are not free. By an appropriate choice of these parameters, we can control the ME. By taking γ = 8, s0 = 0.1, a1 = 0.5, s1 = 0.2. we find the errors are
$\begin{eqnarray*}\begin{array}{l}\{3.9\times {10}^{-6},0.000015625\},\end{array}\end{eqnarray*}$
so the ME is 1.5 × 10−5.
To solve the AEs, we take ${c}_{0}=\tfrac{9{c}_{1}{c}_{2}{c}_{3}-2{c}_{2}^{3}}{27{c}_{3}^{2}}$, and the solutions of (5) and (6) are,
$\begin{eqnarray*}\begin{array}{l}{u}_{1}(x,t)=\displaystyle \frac{{P}_{1}}{{Q}_{1}},\\ {P}_{1}=\left(-3{a}_{1}{c}_{3}{s}_{0}\left({{\rm{e}}}^{2{c}_{1}\left({A}_{0}+z\right)}+{{\rm{e}}}^{\tfrac{2{c}_{2}^{2}\left({A}_{0}+z\right)}{3{c}_{3}}}\right)\right.\\ +{a}_{1}\left({c}_{2}\left(-{s}_{0}\left({{\rm{e}}}^{2{c}_{1}\left({A}_{0}+z\right)}+{{\rm{e}}}^{\tfrac{2{c}_{2}^{2}\left({A}_{0}+z\right)}{3{c}_{3}}}\right)\right)\right.\\ \times \,\left.\sqrt{3{c}_{2}^{2}\left({{\rm{e}}}^{4{c}_{1}\left({A}_{0}+z\right)}+{{\rm{e}}}^{\tfrac{2\left({c}_{2}^{2}+3{c}_{1}{c}_{3}\right)\left({A}_{0}+z\right)}{3{c}_{3}}}\right)-K}+{c}_{2}\right){s}_{1}),\\ K=9{c}_{1}{c}_{3}\left({{\rm{e}}}^{\left(\tfrac{2{c}_{2}^{2}\left({A}_{0}+z\right)}{3{c}_{3}}+2{c}_{1}\left(2{A}_{0}+z\right)\right)}+{{\rm{e}}}^{4{c}_{1}\left({A}_{0}+z\right)}\right),\\ {Q}_{1}={s}_{1}\left(3{c}_{3}{s}_{0}\left({{\rm{e}}}^{2{c}_{1}\left({A}_{0}+z\right)}+{{\rm{e}}}^{\tfrac{2{c}_{2}^{2}\left({A}_{0}+z\right)}{3{c}_{3}}}\right)\right.\\ +\left(\left(-\left({{\rm{e}}}^{2{c}_{1}\left({A}_{0}+z\right)}+{{\rm{e}}}^{\tfrac{2{c}_{2}^{2}\left({A}_{0}+z\right)}{3{c}_{3}}}\right)\right){c}_{2}\right.\\ \left.\left.+\sqrt{3{c}_{2}^{2}\left({{\rm{e}}}^{4{c}_{1}\left({A}_{0}+z\right)}+{{\rm{e}}}^{\tfrac{2\left({c}_{2}^{2}+3{c}_{1}{c}_{3}\right)\left({A}_{0}+z\right)}{3{c}_{3}}}\right)-K}\right){s}_{1}\right),\\ {u}_{2}(x,t)={{vu}}_{1}(x,t),\end{array}\end{eqnarray*}$
$\begin{eqnarray}\begin{array}{l}{u}_{1}(-x,-t)=\displaystyle \frac{{P}_{2}}{{Q}_{2}},\\ {P}_{2}=\left(-3{a}_{1}{c}_{3}{s}_{0}({{\rm{e}}}^{2{c}_{1}\left({A}_{0}-z\right)}+{{\rm{e}}}^{\tfrac{2{c}_{2}^{2}\left({A}_{0}-z\right)}{3{c}_{3}}})\right.\\ +{a}_{1}\left({c}_{2}\left(-{s}_{0}\left({{\rm{e}}}^{2{c}_{1}\left({A}_{0}-z\right)}+{{\rm{e}}}^{\tfrac{2{c}_{2}^{2}\left({A}_{0}-z\right)}{3{c}_{3}}}\right)\right)\right.\\ \times \,\left.\sqrt{3{c}_{2}^{2}\left({{\rm{e}}}^{4{c}_{1}\left({A}_{0}-z\right)}+{{\rm{e}}}^{\tfrac{2\left({c}_{2}^{2}+3{c}_{1}{c}_{3}\right)\left({A}_{0}-z\right)}{3{c}_{3}}}\right)-K}+{c}_{2}\right){s}_{1}),\\ K=9{c}_{1}{c}_{3}\left({{\rm{e}}}^{\left(\tfrac{2{c}_{2}^{2}\left({A}_{0}-z\right)}{3{c}_{3}}+2{c}_{1}\left(2{A}_{0}-z\right)\right)}+{{\rm{e}}}^{4{c}_{1}\left({A}_{0}-z\right)}\right),\\ {Q}_{2}={s}_{1}\left(3{c}_{3}{s}_{0}({{\rm{e}}}^{2{c}_{1}\left({A}_{0}-z\right)}+{{\rm{e}}}^{\tfrac{2{c}_{2}^{2}\left({A}_{0}-z\right)}{3{c}_{3}}})\right.\\ +\left(\left(-\left({{\rm{e}}}^{2{c}_{1}\left({A}_{0}-z\right)}+{{\rm{e}}}^{\tfrac{2{c}_{2}^{2}\left({A}_{0}-z\right)}{3{c}_{3}}}\right)\right){c}_{2}\right.\\ \left.\left.+\sqrt{3{c}_{2}^{2}\left({{\rm{e}}}^{4{c}_{1}\left({A}_{0}-z\right)}+{{\rm{e}}}^{\tfrac{2\left({c}_{2}^{2}+3{c}_{1}{c}_{3}\right)\left({A}_{0}-z\right)}{3{c}_{3}}}\right)-K}\right){s}_{1}\right),\\ {u}_{2}(-x,-t)={{vu}}_{1}(-x,-t),\,z={hx}+{rt}.\end{array}\end{eqnarray}$
By using the results in (20) Rew(x, t) and Rew(−x, −t) are displayed in figures 3(a)–(f).
Figure 3. (a)–(f), Rew(x, t) is displayed in figures 3(a)–(c), while Rew(−x, −t) is displayed in figures 3(d) and (f). When &ohgr; = 5, γ = 2, α = 2, A0 = 0, c1 = 0.5, s2 = 6, s1 = 1.2, r = 0.5, h = 0.3, a2 = 4, a1 = 1.5, c2 = 1.5, a2 = 1.5.
Figures 3(a)–(c) show that the pulses of the field propagate mainly on x > 0, and they exhibit left cable shape, while those of the reverse field propagate mainly on x < 0, showing right cable shapes. Figure 3(a) shows complex oscillatory pulses, while (3(b)) shows complex chirped pulses. Figure 3(d) shows rhombus (diamo0nd ) shape, while 3(e) shows oscillatory pulses.

4. The interactive case

When taking into account the interaction of the field and its reverse field, and in view of section 3, the solutions have to be expressed in terms of g(z) together with g(−z). In the present case the solutions found are exact.

4.1. Exact solution when k = 1 and s = 2

The solutions are written,
$\begin{eqnarray}\begin{array}{l}{U}_{1}(z)=\displaystyle \frac{{a}_{2}g(-z)+{a}_{1}g(z)+{a}_{0}}{{s}_{2}g(-z)+{s}_{1}g(z)+{s}_{0}},\\ {U}_{2}(z)=v\displaystyle \frac{{a}_{2}g(-z)+{a}_{1}g(z)+{a}_{0}}{{s}_{2}g(-z)+{s}_{1}g(z)+{s}_{0}},\\ {U}_{1}(-z)=\displaystyle \frac{{a}_{2}g(z)+{a}_{1}g(-z)+{a}_{0}}{{s}_{2}g(z)+{s}_{1}g(-z)+{s}_{0}},\\ {U}_{2}(-z)=v\displaystyle \frac{{a}_{2}g(z)+{a}_{1}g(-z)+{a}_{0}}{{s}_{2}g(z)+{s}_{1}g(-z)+{s}_{0}},\\ g^{\prime} (z)={c}_{1}g(z)+{c}_{0},\\ g^{\prime} (-z)={c}_{1}g(-z)+{c}_{0}.\end{array}\end{eqnarray}$
By inserting (21) into (8) and (9) leads to,
$\begin{eqnarray}\begin{array}{l}r=\displaystyle \frac{2{\alpha }^{2}{hk}}{\sqrt{4{\alpha }^{2}{\gamma }^{2}{k}^{2}+1}},\ {a}_{1}=-\displaystyle \frac{{s}_{1}{s}_{2}\left(-{\gamma }^{2}{\omega }^{2}+{\alpha }^{2}{k}^{2}+\omega \right)}{{a}_{2}\beta \left({v}^{2}+1\right)},\\ \beta =-\displaystyle \frac{{s}_{2}^{2}\left(-{\gamma }^{2}{\omega }^{2}+{\alpha }^{2}{k}^{2}+\omega \right)}{{a}_{2}^{2}\left({v}^{2}+1\right)},\\ \\ v=\displaystyle \frac{\sqrt{4{\alpha }^{2}{\gamma }^{2}{k}^{2}+1}\left(-\tfrac{{\alpha }^{2}{c}_{1}^{2}{h}^{2}}{4{\alpha }^{2}{\gamma }^{2}{k}^{2}+1}-\tfrac{\left({s}_{1}+{s}_{2}\right)\left(-{\gamma }^{2}{\omega }^{2}+{\alpha }^{2}{k}^{2}+\omega \right)}{{s}_{2}}\right)}{2{\alpha }^{2}{c}_{1}{hk}\left(2{\gamma }^{2}\omega +\sqrt{4{\alpha }^{2}{\gamma }^{2}{k}^{2}+1}-1\right)},\\ {c}_{0}=\displaystyle \frac{{M}_{1}}{{M}_{2}},\\ {M}_{1}={c}_{1}\left(-{a}_{2}{s}_{0}{s}_{1}^{2}\left(4{\alpha }^{2}{\gamma }^{2}{k}^{2}+1\right)\left(-{\gamma }^{2}{\omega }^{2}+{\alpha }^{2}{k}^{2}+\omega \right)\right.\\ {a}_{0}{s}_{2}^{2}\left(2{s}_{1}+{s}_{2}\right)\left(4{\alpha }^{2}{\gamma }^{2}{k}^{2}+1\right)\left(-{\gamma }^{2}{\omega }^{2}+{\alpha }^{2}{k}^{2}+\omega \right)\\ \left.-2{\alpha }^{2}{a}_{2}{c}_{1}^{2}{h}^{2}{s}_{0}{s}_{2}^{2}\right),\\ {M}_{2}={a}_{2}{s}_{2}(({s}_{1}^{2}-{s}_{2}^{2})\left(4{\alpha }^{2}{\gamma }^{2}{k}^{2}+1\right)(-{\gamma }^{2}{\omega }^{2}+{\alpha }^{2}{k}^{2}+\omega )\\ +2{\alpha }^{2}{c}_{1}^{2}{h}^{2}{s}_{2}(2{s}_{2}-{s}_{1})),\\ {c}_{1}=\displaystyle \frac{\sqrt{{s}_{1}-2{s}_{2}}\left({s}_{1}+{s}_{2}\right)\sqrt{\left(4{\alpha }^{2}{\gamma }^{2}{k}^{2}+1\right)\left({\gamma }^{2}{\omega }^{2}+{\alpha }^{2}\left(-{k}^{2}\right)-\omega \right)}}{\sqrt{6}\alpha {{hs}}_{2}^{3/2}}.\end{array}\end{eqnarray}$
The solutions of (5) and (6) are,
$\begin{eqnarray*}\begin{array}{l}{u}_{1}(x,t)=\displaystyle \frac{{P}_{+}}{{}_{Q+}},\\ {P}_{+}={a}_{2}{a}_{0}{s}_{2}\left({s}_{1}^{3}-3{s}_{2}{s}_{1}^{2}-3{s}_{2}^{2}{s}_{1}-2{s}_{2}^{3}\right){H}_{+}\end{array}\end{eqnarray*}$
$\begin{eqnarray}\begin{array}{l}+\left(-{a}_{2}^{2}{s}_{0}\left({s}_{1}^{3}-3{s}_{2}{s}_{1}^{2}-3{s}_{2}^{2}{s}_{1}-2{s}_{2}^{3}\right){H}_{+}\right.\\ \times \,\left.{a}_{2}^{2}+{A}_{0}\left({s}_{1}^{3}-3{s}_{2}{s}_{1}^{2}+3{s}_{2}^{2}{s}_{1}+{s}_{2}^{3}\right)\left({s}_{1}{K}_{+}+{s}_{2}\right)\right),\\ {H}_{\pm }={{\rm{e}}}^{\left(\tfrac{\pm \sqrt{{s}_{1}-2{s}_{2}}\left({s}_{1}+{s}_{2}\right)z\sqrt{-\left(4{\alpha }^{2}{\gamma }^{2}{k}^{2}+1\right)\left(-{\gamma }^{2}{\omega }^{2}+{\alpha }^{2}{k}^{2}+\omega \right)}}{\sqrt{6}\alpha {{hs}}_{2}^{3/2}}\right)},\\ {K}_{\pm }={{\rm{e}}}^{\left(\pm \tfrac{\sqrt{\tfrac{2}{3}}\sqrt{{s}_{1}-2{s}_{2}}\left({s}_{1}+{s}_{2}\right)z\sqrt{-\left(4{\alpha }^{2}{\gamma }^{2}{k}^{2}+1\right)\left(-{\gamma }^{2}{\omega }^{2}+{\alpha }^{2}{k}^{2}+\omega \right)}}{\alpha {{hs}}_{2}^{3/2}}\right)},\\ {Q}_{\pm }=-3{a}_{0}{s}_{2}^{4}\left(2{s}_{1}+{s}_{2}\right){H}_{\pm }+{a}_{2}{s}_{2}\left(3{s}_{0}{s}_{2}^{2}\left(2{s}_{1}+{s}_{2}\right){H}_{\pm }\right.\\ \left.+{A}_{0}\left({s}_{1}^{3}-3{s}_{2}{s}_{1}^{2}+3{s}_{2}^{2}{s}_{1}+{s}_{2}^{3}\right)\left({s}_{1}{K}_{\pm }+{s}_{2}\right)\right),\\ {u}_{1}(-x,-t)=\displaystyle \frac{{P}_{-}}{{Q}_{-}},\\ {P}_{-}={a}_{2}{a}_{0}{s}_{2}\left({s}_{1}^{3}-3{s}_{2}{s}_{1}^{2}-3{s}_{2}^{2}{s}_{1}-2{s}_{2}^{3}\right){H}_{-}\\ +\left(-{a}_{2}^{2}{s}_{0}\left({s}_{1}^{3}-3{s}_{2}{s}_{1}^{2}-3{s}_{2}^{2}{s}_{1}-2{s}_{2}^{3}\right){H}_{-}\right.\\ \left.+{a}_{2}^{2}+{A}_{0}\left({s}_{1}^{3}-3{s}_{2}{s}_{1}^{2}+3{s}_{2}^{2}{s}_{1}+{s}_{2}^{3}\right)\left({s}_{1}{K}_{-}+{s}_{2}\right)\right),\\ {u}_{2}(x,t)={{vu}}_{1}(x,t),\,{u}_{2}(-x,-t)={{vu}}_{1}(-x,-t),\\ z=\displaystyle \frac{2{\alpha }^{2}{hkt}}{\sqrt{4{\alpha }^{2}{\gamma }^{2}{k}^{2}+1}}+{hx}.\end{array}\end{eqnarray}$
We mention that when displaying Rew(x, t) and Rew(−x, −t) in figures, it is observed that they are mainly identical. So, Rew(x, t) is only displayed in figures 4(a) and (b).
Figure 4. (a)–(d) The 3D plot for Rew(x, t) is displayed against x and t. When A0 = 1.5, s1 = 2.5, s2 = 1.2, a1 = 3, α = 1.4, s0 = 1.5, r = 1.3, h = 0.4, &ohgr; = 5, k = 3, a0 = −0.1, a2 = 3.
Figures 4(a) and (b) show that the pulses of the field propagate in the whole space. The effect of varying the hyperbolic coefficient is significant on the field but insignificant change occurs in the reverse field.

4.2. Approximate solution when k = 2 and s = 2

In this case, the solutions are given as in (21), but the AEs take the form,
$\begin{eqnarray}\begin{array}{l}g^{\prime} (z)={c}_{2}g{\left(z\right)}^{2}+{c}_{1}g(z)+{c}_{0},\\ \quad g^{\prime} (-z)={c}_{2}g{\left(-z\right)}^{2}+{c}_{1}g(-z)+{c}_{0}.\end{array}\end{eqnarray}$
From (21) and (24) into (8) and (9) yields,
$\begin{eqnarray}\begin{array}{l}{a}_{1}=\displaystyle \frac{{a}_{2}{s}_{1}}{{s}_{2}},\ \omega =\displaystyle \frac{1-2\alpha \gamma k}{2{\gamma }^{2}},v=\displaystyle \frac{m}{\sqrt{3}},\\ {s}_{0}=-2{a}_{0}\sqrt{\beta }\gamma \sqrt{{m}^{2}-1}\\ {s}_{2}=2{a}_{2}\sqrt{\beta }\gamma \sqrt{{m}^{2}-1},\ m=\displaystyle \frac{\sqrt{4{a}_{2}^{2}\beta {\gamma }^{2}+{s}_{1}^{2}}}{2{a}_{2}\sqrt{\beta }\gamma },\\ {s}_{1}=2\sqrt{2}{a}_{2}\sqrt{\beta }\gamma .\end{array}\end{eqnarray}$
The RT are,
$\begin{eqnarray}\begin{array}{l}\left\{16\sqrt{2}{a}_{0}^{3}{a}_{2}{\beta }^{3/2}\gamma ,-16\sqrt{2}{a}_{0}{a}_{2}^{3}{\beta }^{3/2}\gamma ,\right.\\ 16\sqrt{2}{a}_{0}^{3}{a}_{2}{\beta }^{3/2}\gamma ,-48\sqrt{2}{a}_{0}{a}_{2}^{3}{\beta }^{3/2}\gamma ,\\ -48\sqrt{2}{a}_{0}{a}_{2}^{3}{\beta }^{3/2}\gamma ,-16\sqrt{2}{a}_{0}{a}_{2}^{3}{\beta }^{3/2}\gamma ,\\ -\displaystyle \frac{8\sqrt{2}{a}_{0}^{4}\beta \sqrt{{a}_{2}^{2}\beta {\gamma }^{2}}}{{a}_{2}},8\sqrt{2}{a}_{2}^{3}\beta \sqrt{{a}_{2}^{2}\beta {\gamma }^{2}},\\ 32\sqrt{2}{a}_{2}^{3}\beta \sqrt{{a}_{2}^{2}\beta {\gamma }^{2}},48\sqrt{2}{a}_{2}^{3}\beta \sqrt{{a}_{2}^{2}\beta {\gamma }^{2}},\\ \left.32\sqrt{2}{a}_{2}^{3}\beta \sqrt{{a}_{2}^{2}\beta {\gamma }^{2}},8\sqrt{2}{a}_{2}^{3}\beta \sqrt{{a}_{2}^{2}\beta {\gamma }^{2}}\right\}.\end{array}\end{eqnarray}$
By taking a0 = 0.01, a2 = 0.03, γ = 0.6, β = 0.8, The errors are given in table 1.
Table 1. Errors of the approximate solutions.
2.91436 × 10−7 −2.62292 × 10−6 2.91436 × 10−7 −7.86876 × 10−6
−0.0000786876 −7.86876 × 10−6 −2.62292 × 10−6 −4.85726 × 10−8
3.93438 × 10−6 0.0000157375 0.0000236063 0.0000157375
From table 1, we find that the ME is 2.4 × 10−5.
The solutions of (5) and (6) are,
$\begin{eqnarray}\begin{array}{l}{u}_{1}(x,t)=\displaystyle \frac{{P}_{1}}{{Q}_{1}},\quad {P}_{1}={a}_{2}(\sqrt{{c}_{1}^{2}-4{c}_{0}{c}_{2}}{s}_{2}-2{c}_{2}{s}_{0}K),\\ {Q}_{1}=2\sqrt{{m}^{2}-1}\sqrt{\beta }\gamma ({a}_{2}\sqrt{{c}_{1}^{2}-4{c}_{0}{c}_{2}}+2{a}_{0}{c}_{2}K),\\ {u}_{1}(-x,-t)=\displaystyle \frac{{P}_{2}}{{Q}_{2}},\\ {P}_{2}={a}_{2}(-\sqrt{{c}_{1}^{2}-4{c}_{0}{c}_{2}}-2{a}_{0}{c}_{2}K)K),\\ {Q}_{2}=2\sqrt{{m}^{2}-1}\sqrt{\beta }\gamma \left(2{a}_{0}{c}_{2}-{a}_{2}\sqrt{{c}_{1}^{2}-4{c}_{0}{c}_{2}}\right.K),\\ K=\tanh \left(\displaystyle \frac{1}{2}\sqrt{{c}_{1}^{2}-4{c}_{0}{c}_{2}}\left({A}_{0}-z\right)\right)\\ \quad -\tanh \left(\displaystyle \frac{1}{2}\sqrt{{c}_{1}^{2}-4{c}_{0}{c}_{2}}\left({A}_{0}+z\right)\right),\\ {u}_{2}(x,t)={{vu}}_{1}(x,t),\\ {u}_{2}(-x,-t)={{vu}}_{1}(-x,-t),\,z={rt}+{hx}.\end{array}\end{eqnarray}$
The results in (27) are used to display Rew(x, t) and Rew(−x, −t) in figures 5(a)–(d)
Figure 5. Rew(x, t) is displayed in figures 5(a) , while Rew(−x, −t) is displayed in figure 5(b) When &ohgr; = 5, γ = 2, α = 2, A0 = 0, c1 = 0.5, s2 = 6, s1 = 1.2, r = 0.5, h = 0.3, a2 = 4, a1 = 1.5, c2 = 1.5.
Figure 5(a) shows that the pulses of the field propagate in the whole space and it exhibits left cable shape, while figure 5(b) shows that the reverse field exhibits right cable shape with pulses compression.

5. Analysis of modulation instability

The study of MS is considered when a system is governed by an equation that admits normal mode solution (NMS) (plane wave solution). A systems with complex equations possesses NMS, so the study of MS holds. In the present case, (2) has the solutions,
$\begin{eqnarray}\begin{array}{l}w(x,t)=A{{\rm{e}}}^{{\rm{i}}({Kx}-t{\rm{\Omega }})},\\ w(-x,-t)=A{{\rm{e}}}^{-{\rm{i}}({Kx}-t{\rm{\Omega }})},A\gt 0,\\ {\rm{\Omega }}=\displaystyle \frac{1\pm \sqrt{4{A}^{2}\beta {\gamma }^{2}+4{\alpha }^{2}{\gamma }^{2}{K}^{2}+1}}{2{\gamma }^{2}}.\end{array}\end{eqnarray}$
We use the perturbation expansion,
$\begin{eqnarray}\begin{array}{l}w(x,t)={{\rm{e}}}^{{\rm{i}}({Kx}-t{\rm{\Omega }})}\\ \,\times \left(A+{{\rm{e}}}^{\lambda t}({\epsilon }_{1}{U}_{1}(x)+{\rm{i}}{\epsilon }_{2}{U}_{2}(x))+O{\left({\varepsilon }_{i}\right)}^{2}\right),\\ w(-x,-t)={{\rm{e}}}^{-{\rm{i}}({Kx}-t{\rm{\Omega }})}\\ \,\times \left(A+{{\rm{e}}}^{-\lambda t}({\epsilon }_{1}{U}_{1}(-x)+{\rm{i}}{\epsilon }_{2}{U}_{2}(-x))+O{\left({\varepsilon }_{i}\right)}^{2}\right),i=1,2.\end{array}\end{eqnarray}$
By inserting (28) into (2), we get,
$\begin{eqnarray}\begin{array}{rcl}M\left(\begin{array}{c}{\epsilon }_{1}\\ {\epsilon }_{2}\end{array}\right) & = & 0,\,M=\left(\begin{array}{cc}{m}_{11} & {m}_{12}\\ {m}_{21} & {m}_{22}\end{array}\right),\\ {m}_{11} & = & {A}^{2}\beta {U}_{1}(-x)+{A}^{2}\beta {U}_{1}(x)+{\gamma }^{2}{\lambda }^{2}{U}_{1}(x),\\ {m}_{12} & = & \lambda {U}_{1}(x)\left(-\sqrt{4{A}^{2}\beta {\gamma }^{2}+4{\alpha }^{2}{\gamma }^{2}{K}^{2}+1}\right),\\ {m}_{21} & = & \lambda {U}_{2}(x)\sqrt[4]{{\left(4{A}^{2}\beta {\gamma }^{2}+4{\alpha }^{2}{\gamma }^{2}{K}^{2}+1\right)}^{2}}\\ & & +\,2{\alpha }^{2}{{KU}}_{2}^{{\rm{{\prime} }}}(x),\\ {m}_{22} & = & {A}^{2}\beta {U}_{2}(-x)+{A}^{2}\beta {U}_{2}(x)-{\alpha }^{2}{U}_{2}^{{\rm{{\prime} }}{\rm{{\prime} }}}(x)\\ & & +\,{\gamma }^{2}{\lambda }^{2}{U}_{2}(x).\end{array}\end{eqnarray}$
The solution of (29) is ${\det }(M)=0,$ which leads to,
$\begin{aligned}& A^{2}\beta U_{1}(-x)(U_{2}(x)(A^{2}\beta+\gamma^{2}\lambda^{2}) \\&+A^2\beta U_2(-x)-\alpha^2U_2''(x)) \\&+U_1(x)(A^2\beta U_2(-x)(A^2\beta+\gamma^2\lambda^2) \\&+(A^4\beta^2+2A^2\beta\gamma^2\lambda^2\lambda^2+4A^2\beta\gamma^2+4\alpha^2\gamma^2K^2+1+\gamma^4\lambda^4) \\&\times U_2(x)+\alpha^2(2\lambda K U_2'(x) \\&\times\sqrt{4A^2\beta\gamma^2+4\alpha^2\gamma^2K^2+1}-U_2''(x)(A^2\beta+\gamma^2\lambda^2))).\end{aligned}$
The solutions of the eigenvalue problem (30) is subjected to the boundary conditions ∣Uix)∣ ≤ Hi, i = 1, 2. This suggests
$\begin{eqnarray}{U}_{i}(\pm x)={H}_{i}{{\rm{e}}}^{\pm {\rm{i}}{ax}}+C.C.,i=1,2,\end{eqnarray}$
where C. C. is the complex conjugate. Calculations give rise to,
$\begin{eqnarray}\begin{array}{l}\beta =\displaystyle \frac{1}{25}\left(-\displaystyle \frac{5}{{A}^{2}{\gamma }^{2}}-\displaystyle \frac{4{\alpha }^{2}{K}^{2}}{{A}^{2}}\pm \displaystyle \frac{2\sqrt{2}\sqrt{52{\alpha }^{4}{\gamma }^{2}{K}^{4}+5{\alpha }^{2}{K}^{2}}}{{A}^{2}\gamma }\right),\\ a=\displaystyle \frac{\pm \sqrt{2}\gamma \lambda }{\alpha },\\ \lambda =\displaystyle \frac{\pm {\left(4{A}^{2}\beta {\gamma }^{2}+4{\alpha }^{2}{\gamma }^{2}{K}^{2}+1\right)}^{1/4}\sqrt{2\sqrt{2}\alpha \gamma K+\sqrt{4{A}^{2}\beta {\gamma }^{2}+4{\alpha }^{2}{\gamma }^{2}{K}^{2}+1}}}{{\gamma }^{2}}.\end{array}\end{eqnarray}$
In (32), the plus sign may be taken to correspond the field, while the minus sign corresponds to the reverse field (or vice versa). We find that the field is modulationally unstable while the reverse field is modulationally stable (or vice versa). For β < 0, (or β > 0) the medium is self-defocusing (or self- focusing) polarization. It is worth mentioning that the solutions found in sections 3 and 4 hold when β < 0, so the case when β > 0 is not applicable.

6. Conclusions

The study of wave-operator nonlinear Schrödinger equation with space–time reverse in two cases was carried out. In the first case when the field and its reverse are not interactive and in the second case when they are interactive. In both two cases, the solutions of the aforementioned equation are found by implementing the unified method together with conventional formulation of solutions. In the two cases, exact and approximate analytic solutions are obtained. For approximate solutions, the maximum error is controlled via an appropriate choice of the parameters in the residue terms. It is worth mentioning that the error is space and time independent, which reveals that there is no limitation for the validity of solutions. This is not the case in solutions evaluated via numerical techniques. The solutions are represented graphically. In the non-interactive case, it is observed that the field shows solitons propagating on the left-half space while the reverse filed solitons propagate in the right-half space (or vice versa). This behavior was not observed in the research works in the literature. We think that this property is a main characteristic of complex field systems with space-time reverse. In the interactive, the field and its reverse solitons propagate in the whole space, but with low density in the right-half. When analyzing the stability of normal modes, it is found that the field and its reverse are stable for a negative Kerr nonlinearity coefficient. It is worth mentioning that the solutions obtained here, verify this property.

Declarations

Ethical approval

The author declares that there are no animal studies in this work.

Competing interests

The author declares that there is no conflict of interest.

Authors’ contributions

Wrote the paper, conceptualization, and revision.

Funding

No funding available.

1
Ablowitz M J Feng B F Luo X-D Musslimani Z H 2018 Inverse scattering transform for the nonlocal reverse space–time nonlinear Schrödinger equation Theor. Math. Phys. 196 1241 1267

DOI

2
Ablowitz M J Feng B-F Luo X-D Musslimani Z H 2017 Inverse scattering transform for the nonlocal reverse space–time sine-Gordon, sinh-Gordon, and nonlinear Schrödinger equations with nonzero boundary conditions arXiv:1703.02226v1 [math-ph]

3
Ablowitz M J Feng B-F Luo X-D Musslimani Z H 2018 Reverse space–time nonlocal Sine-Gordon/Sinh-Gordon equations with nonzero boundary conditions Stud. Appl. Math. 141 267 307

DOI

4
Ma W-X 2021 Inverse scattering and soliton solutions of nonlocal reverse-spacetime nonlinear Schrödinger equations Proc. Amer. Math. Soc. 149 251 263

DOI

5
Song C Xiao D Zhu Z-N 2017 Reverse space–time nonlocal Sasa–Satsuma equation and its solutions J. Phys. Soc. Jpn. 86 054001

DOI

6
Ma W-X 2020 Inverse scattering and soliton solutions of nonlocal complex reverse-spacetime mKdV equations J. Geom. and Phys. 57 103845

DOI

7
Liu W Zheng X Li X 2018 X., Bright and dark soliton solutions to the partial reverse space–time nonlocal Mel’nikov equation Nonlinear Dyn. 94 2177 2189

DOI

8
Mosammam A M 2015 The reverse dimple in potentially negative-value space–time covariance models Stoch. Environ. Res. Risk Assess. 29 599 607

DOI

9
Song J Y Xiao Y Zhang C P 2022 Darboux transformation, exact solutions and conservation laws for the reverse space–time Fokas–Lenells equation Nonlinear Dyn. 107 3805 3818

DOI

10
Sarfraz H Hanif Y Saleem U 2020 Novel solutions of general and reverse space–time nonlocal coupled integrable dispersionless systems Res. Phys. 16 102893

DOI

11
Zhang W-X Liu Y 2022 Integrability and multisoliton solutions of the reverse space and/or time nonlocal Fokas–Lenells equation Nonlinear Dyn. 108 2351 2349

DOI

12
Luoa X-D 2019 Inverse scattering transform for the complex reverse space-time nonlocal modified Korteweg-de Vries equation with nonzero boundary conditions and constant phase shift Chaos 29 073118

DOI

13
Zhou H Chen Y 2021 Breathers and rogue waves on the double-periodic background for the reverse-space–time derivative nonlinear Schrödinger equation Nonlinear Dyn. 106 3437 3451

DOI

14
Ling L Ma W-X 2021 Inverse scattering and soliton solutions of nonlocal complex reverse-space–time modified Korteweg-de Vries hierarchies Symmetry 13 512

DOI

15
Liu W Qin Z Chow K W Lou S 2020 Families of rational and semirational solutions of the partial reverse space–time nonlocal Mel’nikov equation Complexity 2020 2642654

DOI

16
Saenger E H Kocur G K Jud R Torrilhon M 2011 Application of time reverse modeling on ultrasonic non-destructive testing of concrete Appl. Math. Model. 35 807 816

DOI

17
Cai-Qin S Zuo-Nong Z 2020 An integrable reverse space–time nonlocal Sasa–Satsuma equation Acta Phys. Sin. 69 20191887

DOI

18
Wirth J 2002 On the existence of the Moller wave operator for wave equations with small dissipative terms arXiv:math/0210098

19
Durand P 1983 direct determination of effective Hamiltonians by wave-operator methods. I. General formalism Phys. Rev. A 28 3184

DOI

20
Ting-chun W Lu-ming Z 2006 Analysis of some new conservative schemes for nonlinear Schrödinger equation with wave operator Appl. Math. Comput. 182 1780 1794

DOI

21
Bao W Cai Y 2012 Uniform error estimates of finite difference methods for the nonlinear Schrödinger equation with wave operator SIAM J. Numer. Anal. 50 1

DOI

22
Cai W He D Pan K 2019 A linearized energy-conservative finite element method for the nonlinear Schrödinger equation with wave operator Appl. Numer. Math. 140 183 198

DOI

23
Li X Zhang L Wang S 2012 A compact finite difference scheme for the nonlinear Schrödinger equation with wave operator Appl. Math. Comput. 219 3187 3197

DOI

24
Guo L Xu Y 2015 Energy conserving local discontinuous Galerkin methods for the nonlinear Schrödinger equation with wave operator J. Sci. Comput. 65 622 647

DOI

25
Bao W Cai Y 2014 Uniform and optimal error estimates of an exponential wave integrator sine pseudospectral method for the nonlinear Schrödinger equation with wave operator SIAM J. Numer. Anal. 52 1103 1496

DOI

26
Zhang L Chang Q 2003 A conservative numerical scheme for a class of nonlinear Schrödinger equation with wave operator Appl. Math. Comput. 145 603 612

DOI

27
Lia M Zhao Y-L 2018 A fast energy conserving finite element method for the nonlinear fractional Schrödinger equation with wave operator Appl. Math. Comput. 338 758 773

DOI

28
Shimomura A 2003 Modified wave operator for the coupled wave Schrödinger equation in three space dimensions Disc. Cont. Dyn. Syst. 9 1571 1586

DOI

29
Brugnano L Zhang C Li D 2018 A class of energy-conserving Hamiltonian boundary value methods for nonlinear Schrödinger equation with wave operator Communi. Nonl. Sci. Numer. Simul. 60 33 49

DOI

30
Guo L Xu Y 2015 Energy conserving local discontinuous Galerkin methods for the nonlinear Schrödinger equation with wave operator J. Sci. Comput. 65 622 647

DOI

31
Saha S Singh R S 2021 New various multisoliton kink-type solutions of the (1+1)-dimensional Mikhailov–Novikov–Wang equation Math Meth. Appl. Sci. 44 14690 11470

DOI

32
Singh S Ray S S 2021 Painlevé analysis, auto-Bäcklund transformation and analytic solutions for modified KdV equation with variable coefficients describing dust acoustic solitary structures in magnetized dusty plasma Mod. Phys. Lett. B 35 2150464

DOI

33
Singh S Ray S S 2022 New abundant analytic solutions for generalized KdV6 equation with time-dependent variable coefficients using Painlevé analysis and auto-Bäcklund transformation Int. J. Geom. Meth. Mod. Phys. 19 2250086

DOI

34
Ray S S Singh S 2022 New bright soliton solutions for Kadomtsev–Petviashvili–Benjamin–Bona–Mahony equations and bidirectional propagation of water wave surface Int. J. Mod. Phys. C 33 2250069

DOI

35
Singh S Ray S S Painlevé integrability, auto-Bäcklund transformations, new abundant analytic solutions including multi-soliton solutions for time-dependent extended KdV8 equation in nonlinear physics J. Ocean Eng. Sci. doi: 10.1016/j.joes.2022.03.020

36
Wang X-B Tian S-F 2022 Exotic vector freak waves in the nonlocal nonlinear Schrödinger equation Physica D: Nonl. Pheno. 442 133528

DOI

37
Wang X-B Tian S-F 2022 Exotic localized waves in the shifted nonlocal multicomponent nonlinear Schrödinger equation Theor. Math Phys. 212 1193 1210

DOI

38
Ma W-X 2023 Sasa–Satsuma type matrix integrable hierarchies and their Riemann–Hilbert problems and soliton solutions Physica D: Nonl. Pheno. 446 133672

DOI

39
Ma W-X 2022 Reduced nonlocal integrable mKdV equations of type (−λ, λ) and their exact soliton solutions Commun. Theor. Phys. 74 065002

DOI

40
Abdel-Gawad H I 2012 Towards a unified method for exact solutions of evolution equations. An application to reaction diffusion equations with finite memory transport J. Stat. Phys. 147 506 521

DOI

41
Abdel-Gawad H I 2022 Inelastic soliton interactions for nonlinear directional couplers in optical metamaterials with Kerr nonlinearity modulation stability J. Nonl. Opt. Phys. Mater. 31 2250016

DOI

42
Abdel-Gawad H I Tantawy M Fahmy E S Park C 2022 Langmuir waves trapping in a (1+2) dimensional plasma system. Spectral and modulation stability analysis Chinese J. Phys. 77 2148 2159

DOI

43
Abdel-Gawad H I 2022 Intricate and multiple chirped waves geometric structures solutions of two-mode KdV equation, spectral and stability analysis Int. J. Mod. Phys. B 36 2250056

DOI

44
Abdel-Gawad H I 2022 Continuum soliton chain analog to Heisenberg spin chain system. Modulation stability and spectral characteristics Int. J. Theo. Phys. 61 188

DOI

45
Abdel-Gawad H I 2022 Infinite solitons in ferromagnetic materials with an internal magnetic field Mod. Phys. Lett. B 35 2150413

DOI

46
Wazwaz A M 2004 The tanh method for traveling wave solutions of nonlinear equations Appl. Math. Comput. 154 713 723

DOI

47
He J H Wu X H 2006 Exp−function method for nonlinear wave equations Chaos, Solitons Fractals 30 700 708

DOI

48
Bekir A 2008 Application of the expansion method for nonlinear evolution equations Phys. Lett. A 372 3400 3406

DOI

49
Bueno M I Marcellán F 2004 Darboux transformation and perturbation of linear functionals Lin. Algebra Appl. 384 215 242

DOI

50
Hosseini K R Ansari R 2017 New exact solutions of nonlinear conformable time−fractional Boussinesq equations using the modified Kudryashov method Waves Rand. Compl. Media 27 628

DOI

51
Abdel-Gawad H I 2023 Dynamics of steady, unsteady flows and heat transfer in Casson fluid over a free stretching surface: stability analysis Waves Rand. Compl. Media doi: 10.1080/17455030.2023.2176171

52
Abdel-Gawad H I 2023 Approximate-analytic optical soliton solutions of a modified-Gerdjikov–Ivanov equation: modulation instability Opt. Quant. Electron. 55 298

DOI

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